Package {gcloglog}

Create a Generalized complementary log-log link-function for GLM-type family

Description

This function is used with "family" functions in "glm" type modeling functions.

Usage

make.gcloglog(phi)

Arguments

Value

An object of class "link-glm, a list with (amongst others) the components:

linkfun:

Link function function(mu) for fixed phi

linkinv:

Inverse link function function(eta) for fixed phi

variance:

variance function

mu.eta:

Derivative function(eta)

valideta:

function(eta)TRUE

validmu:

function(mu) that ensures 0<p<1

name:

the name for the link gcloglog(phi)

Author(s)

Bert van der Veen

References

van der Veen and Hui (2025). In prep.

Examples

# Define logit link via gcloglog
library(gcloglog)
gcloglog1 <- make.gcloglog(1)

# Fit logistic regression with gcloglog
x <- rnorm(100)
b1 = 2.3
b0 = 0.1
eta = b0+b1*x
y = rbinom(length(x), 1, plogis(eta))
data <- data.frame(y = y,x = x)
model <- glm(y~x, family = binomial(link = gcloglog1), data = data)

MLE estimation of the shape of a binomial link function

Description

This function takes a model object, estimates a parameter phi that controls the shape of the link function, and performs further profiling for retrieving a confidence interval. The implementation heavily borrows from tmbprofile in the TMB package.

Usage

## S3 method for class 'gcloglog'
profile(
  model,
  CI = TRUE,
  alpha = 0.05,
  plot = TRUE,
  h = 0.02,
  ytol = 2,
  ystep = 0.1,
  maxit = 100,
  adaptive = TRUE,
  trace = TRUE,
  warmstart = TRUE,
  ...
)

Arguments

Details

The final model inherits from class "gcloglog", so that its vcov, summary, and confint return standard errors that account for the estimation of phi; vcov(..., correct = FALSE) and summary(..., correct = FALSE) give those conditional on phi. confint defaults to Wald confidence intervals from the corrected covariance; method = "profile" (and "boot" for glmer fits) are other options conditional on phi.

For a glm, the corrected asymptotic covariance is the naive covariance conditional on the estimated link shape, plus a rank-one update, which follows from the joint information of the coefficients and log(phi). For a glmer fit, it is the fixed-effect block of the inverse of the joint Hessian of the Laplace-approximated log-likelihood in the random-effect parameters (lme4's theta), the fixed effects, and log(phi), obtained by finite differences (hessian). Random-effect parameters at their boundary (a singular fit) are held fixed.

The correction does not apply when the estimate of phi is at the boundary phi = 0, where the fit is equivalent to cloglog regression (detected as the fit not improving on the log-likelihood of the cloglog link at the same parameter estimates by more than 1e-5), or when the joint information is not positive definite. In those cases the standard errors are conditional on phi, with a warning from vcov and confint, and a note in the printed summary.

Value

A list including the optimisation results, CI, and final model fit.

Author(s)

Bert van der Veen

References

van der Veen and Hui (2025). In prep.

Examples

# Define logit link via gcloglog
library(gcloglog)
gcloglog1 <- make.gcloglog(1)

# Profile the gcloglog link
x <- rnorm(100)
b1 = 2.3
b0 = 0.1
eta = b0+b1*x
y = rbinom(length(x), 1, plogis(eta))
data <- data.frame(y = y,x = x)
model <- glm(y~x, family = binomial(link = gcloglog1), data = data)
res <- profile.gcloglog(model)

final.model <- res$final.model
summary(final.model)                 # standard errors account for the estimate of phi
confint(final.model)
summary(final.model)                 # conditional on the estimate of phi
vcov(final.model, correct = FALSE)   # conditional on the estimate of phi

MLE estimation of the shape of a binomial link function

Description

This function takes a model object with a family argument, and estimates a parameter phi that controls the shape of the link function.

Usage

## S3 method for class 'glm'
profile.phi(
  model,
  y,
  optimizer = nlminb,
  optControl = list(iter.max = 500, eval.max = 500),
  optArgs = list(lower = log(1.400882e-08)),
  warmstart = TRUE,
  ...
)

## S3 method for class 'merMod'
profile.phi(
  model,
  y,
  optimizer = bobyqa,
  optControl = list(maxit = 500),
  optArgs = NULL,
  warmstart = TRUE,
  ...
)

## Default S3 method:
profile.phi(
  model,
  y,
  optimizer = bobyqa,
  optControl = list(maxit = 500),
  optArgs = NULL,
  ...
)

profile.phi(model, ...)

Arguments

Value

The optimisation results of the profiling, with log(phi) as the estimated parameter.

Author(s)

Bert van der Veen

References

van der Veen and Hui (2025). In prep.

Examples

# Define logit link via gcloglog
library(gcloglog)
gcloglog1 <- make.gcloglog(1)

# Fit logistic regression with gcloglog
x <- rnorm(100)
b1 = 2.3
b0 = 0.1
eta = b0+b1*x
y = rbinom(length(x), 1, plogis(eta))
data <- data.frame(y = y,x = x)
model <- glm(y~x, family = binomial(link = gcloglog1), data = data)

# Find optimal link shape by profiling phi
res <- profile.phi(model, y = model.response(model.frame(model)))
exp(res$optr$par) # extract phi